If John's preferences are given by U(c,D)=log(c)+D where c is consumption and D is total donations and a wealthy individual sets up a matching fund campaign where John's total donation (D) is given by D=kd where k is the expansion factor and d is John's personal contribution and his budget constraint is c+d=m.
A) Find John's total donation function D* and his consumption demand function c*
B) What are John's private contributions, d*
C) In the absence of a matching funds campaign what is John's private donation, d*? How much Does John devote to consumption c*? What is the donation D*?
To find John's total donation function D* and his consumption demand function c*, we need to optimize his utility subject to his budget constraint.
A) Optimization problem:
Maximize U(c, D) = log(c) + D
Subject to c + d = m
To solve this problem, we can use Lagrange multipliers. The Lagrangian function is:
L(c, D, λ) = log(c) + D + λ(m - c - d)
Taking the partial derivatives and setting them equal to zero, we have:
∂L/∂c = 1 / c - λ = 0
∂L/∂D = 1 - λk = 0
∂L/∂λ = m - c - d = 0
From the first equation, we can solve for c:
1 / c = λ => c = 1 / λ
Substituting this into the third equation, we get:
m - (1 / λ) - d = 0 => (1 / λ) + d = m => d = m - (1 / λ)
Finally, substituting the value of λ back into c gives us the consumption demand function:
c* = 1 / λ
Now, let's solve for λ:
From the second equation, we have:
1 - λk = 0 => λ = 1 / k
Substituting this into the expression for d, we obtain:
d* = m - (1 / λ) => d* = m - k
B) John's private contribution (d*) is given by m - k.
C) In the absence of a matching funds campaign, k = 0 and thus d* = m. John's private donation is equal to his total budget, d*. The amount John devotes to consumption (c*) is zero since he does not allocate any part of his budget to consumption when there is no matching funds campaign.